Diagonal Strichartz conjecture for orthonormal systems of wave equations

Let $n\geq2$, $q\geq \frac{2(n+1)}{n-1}$, and let $(f_j)_j$ be an orthonormal system in the homogeneous Sobolev space $\dot{H}^s(\mathbb{R}^n)$. For a sequence of coefficients $(\lambda_j)_j$, consider the wave evolution $U_t f_j$. Diagonal Strichartz conjecture. The estimate

$$\Big\| \sum_j \lambda_j |U_t f_j|^2\Big\|_{L_{x,t}^{\frac q2}(\mathbb{R}^{n+1})} \lesssim \|\lambda\|_{\ell^\beta}$$

should hold if $\beta\leq \frac{n}{2(n+1)}q$. The sharp characterization of Strichartz estimates for orthonormal systems of wave equations remains open, and this estimate is presented as a conjectured diagonal case motivated by broader conjectural statements in the literature.